The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations, and shows that the same behavior is attained in the case of non-classical, generalized solutions.

中文翻译：

---

非线性波动方程广义解的奇异性传播

本文致力于非线性小的半线性波动方程广义解的正则性理论。该设置是广义函数的Colombeau代数之一。如图所示，在线性空间中，在一个空间维度上，原点处的初始奇点沿着从原点发出的特征线传播。该证明是基于具有所需正则性的Colombeau解决方案子集上的合适超度量拓扑中的不动点定理。本文采用了1970年代对半线性双曲型方程经典解中的奇异性的初步研究，并表明在非经典广义解的情况下也能达到相同的性能。